Finite Group Databasebeta

← #37554830

Subgroups — #37554830

Order 23488.

334 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19154576C211yesmaximal, characteristic, complemented
2#26044248C211yesmaximal, characteristic, complemented
3#26044242C212yesmaximal, complemented
4#26044240C211yesmaximal, characteristic, complemented
5#26044233C212yesmaximal, complemented
6#26044232C211yesmaximal, characteristic, complemented
7#26044230C212yesmaximal, complemented
8D4×Dic367C211yesmaximal, characteristic, complemented
9C4×C22⋊D367C211yesmaximal, characteristic, complemented
10C2×C22⋊Dic367C212yesmaximal, complemented
11C22×D4×C367C211yesmaximal, characteristic, complemented
12#19154576—24no—
13C2×Dic734C2211yescharacteristic, complemented
14C2×C22⋊D367C2211yescharacteristic
15C2×C22⋊D367C2212yescomplemented
16C2×C22⋊D367C2212yescomplemented
17C22×Dic367C2212yescomplemented
18C22×Dic367C2212yescomplemented
19C2×C4×D367C2211yescharacteristic, complemented
20C4⋊Dic367C2211yescharacteristic, complemented
21Dic367⋊C4C2211yescharacteristic, complemented
22Dic367⋊C4C2212yescomplemented
23C22⋊Dic367C2211yescharacteristic
24C22⋊Dic367C2212yes—
25C22⋊Dic367C2212yescomplemented
26C22⋊Dic367C2212yescomplemented
27C22⋊Dic367C2212yescomplemented
28C22⋊Dic367C2212yescomplemented
29(C2×D367)⋊C4C2211yescharacteristic, complemented
30C4×Dic367C2211yescharacteristic, complemented
31C24×C367C2212yes—
32C2×D4×C367—24no—
33C2×D4×C367C2211yescharacteristic
34C2×D4×C367C2212yes—
35C2×D4×C367C2211yescharacteristic
36C22×C4×C367C2211yescharacteristic
37C22×D367C2311yescharacteristic
38Dic734—22no—
39C22⋊D367—24no—
40C22⋊D367—22no—
41C22⋊D367—42no—
42C2×Dic367—24no—
43C2×Dic367—42no—
44C2×Dic367C2311yescharacteristic
45C2×Dic367C2311yescharacteristic
46C2×Dic367C2312yes—
47C2×Dic367C2311yescharacteristic
48C2×Dic367C2312yes—
49C4×D367—22no—
50C23×C367—24no—
51C23×C367—24no—
52C23×C367—42no—
53C23×C367C2312yes—
54C23×C367C2312yes—
55C23×C367C2311yescharacteristic
56D4×C367—24no—
57D4×C367—22no—
58D4×C367D414yescomplemented
59C2×C4×C367—22no—
60C2×C4×C367C2311yescharacteristic
61C2×C4×C367C2311yescharacteristic
62C22×C367—21no—
63C22×C367—22no—
64C22×C367—21no—
65C22×C367—22no—
66C22×C367—22no—
67C22×C367—22no—
68C22×C367—22no—
69C22×C367—48no—
70C22×C367C2×D414yes—
71C22×C367#4112yescomplemented
72C22×C367C2411yescharacteristic
73C2×D367—21no—
74C2×D367—21no—
75C2×D367—21no—
76C4×C367—21no—
77C4×C367C2×D412yes—
78Dic367—21no—
79Dic367—22no—
80Dic367—21no—
81Dic367—21no—
82Dic367—22no—
83C2×C367—22no—
84C2×C367—24no—
85C2×C367—42no—
86C2×C367C22×D411yescharacteristic
87C2×C367C2×#4111yescharacteristic
88C2×C367H(5,𝔽2)11yescharacteristic
89D367—41no—
90C367#54611yescharacteristic, complemented, Sylow
91#546—3671nomaximal, Sylow
92C2×#41—3671no—
93C22×D4D36711yescharacteristic, complemented
94#124—3671no—
95#123—3672no—
96C22⋊Q8—3671no—
97#121—3671no—
98#121—3672no—
99#120—3672no—
100C4×D4—3671no—
101C4×D4—3671no—
102C2×C22⋊C4—3672no—
103C24C2×D36712yes—
104#41—7344no—
105C2×Q8—3671no—
106C2×D4—24no—
107C2×D4—3672no—
108C2×D4—3671no—
109C2×D4—3672no—
110C2×D4C2×D36712yes—
111C2×D4C2×D36711yescharacteristic
112C2×D4C2×D36711yescharacteristic
113C22×C4—3672no—
114C22×C4—3672no—
115C22×C4—3671no—
116C22×C4C2×D36711yescharacteristic
117C4⋊C4—3671no—
118C4⋊C4—3671no—
119C4⋊C4—3672no—
120C22⋊C4—3671no—
121C22⋊C4—3672no—
122C22⋊C4—3672no—
123C22⋊C4—3672no—
124C22⋊C4—3672no—
125C22⋊C4—3672no—
126C22⋊C4—3671no—
127C42—3671no—
128C23—24no—
129C23—24no—
130C23—42no—
131C23—3671no—
132C23C22×D36712yes—
133C23C22×D36712yes—
134C23C22×D36711yescharacteristic
135Q8—7342no—
136D4—24no—
137D4—22no—
138D4—14682no—
139D4—7344no—
140D4—7342no—
141D4C22⋊D36714yescomplemented
142C2×C4—22no—
143C2×C4—3671no—
144C2×C4—3671no—
145C2×C4—3672no—
146C2×C4—3671no—
147C2×C4—3672no—
148C2×C4—14682no—
149C2×C4—7344no—
150C2×C4—7342no—
151C2×C4C22×D36711yescharacteristic
152C2×C4C22×D36711yescharacteristic
153C22—21no—
154C22—22no—
155C22—22no—
156C22—21no—
157C22—22no—
158C22—22no—
159C22—22no—
160C22—48no—
161C22—7341no—
162C22—7341no—
163C22—7341no—
164C22C2×C22⋊D36714yes—
165C22#1915457612yescomplemented
166C22C23×D36711yescharacteristic
167C4—21no—
168C4—7341no—
169C4—7342no—
170C4—7341no—
171C4—7341no—
172C4—7342no—
173C4C2×C22⋊D36712yes—
174C2—22no—
175C2—24no—
176C2—42no—
177C2—14681no—
178C2C22×C22⋊D36711yescharacteristic
179C2C2×#1915457611yescharacteristic
180C2#2604430611yescharacteristic
subgroupG/H
center, FrattiniC22C23×D367
commutator, socle, cosocleC22×C367C24
FittingC22×D4×C367C2
hypercenterC22×D4D367
hypocenterC367#546
LCSG ▷ C22×C367 ▷ C367
UCSC1 ◁ C22 ◁ C22×D4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#19154576C2semidirect
2. #26044248C2semidirect
3. #26044242C2semidirect
4. #26044240C2semidirect
5. #26044233C2semidirect
6. #26044232C2semidirect
7. #26044230C2semidirect
8. D4×Dic367C2semidirect
9. C4×C22⋊D367C2semidirect
10. C2×C22⋊Dic367C2semidirect
11. C22×D4×C367C2semidirect
13. C2×Dic734C22semidirect
15. C2×C22⋊D367C22semidirect
16. C2×C22⋊D367C22semidirect
17. C22×Dic367C22semidirect
18. C22×Dic367C22semidirect
19. C2×C4×D367C22semidirect
20. C4⋊Dic367C22semidirect
21. Dic367⋊C4C22semidirect
22. Dic367⋊C4C22semidirect
25. C22⋊Dic367C22semidirect
26. C22⋊Dic367C22semidirect
27. C22⋊Dic367C22semidirect
28. C22⋊Dic367C22semidirect
29. (C2×D367)⋊C4C22semidirect
30. C4×Dic367C22semidirect
58. D4×C367D4semidirect
71. C22×C367#41semidirect
90. C367#546semidirect
93. C22×D4D367semidirect
141. D4C22⋊D367semidirect
165. C22#19154576semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#191545762
#260442482
#260442422
#260442402
#260442332
#260442322
#260442302
D4×Dic3672
C4×C22⋊D3672
C2×C22⋊Dic3672
C22×D4×C3672
#546367